# Time-weighted Return

On January 1, Jonathan Wood invests \$50,000. At the end of March, his investment is worth \$51,000. On April 1, Wood deposits \$10,000 into his account, and by the end of June, his account is worth \$60,000. Wood withdraws \$30,000 on July 1 and makes no additional deposits or withdrawals the rest of the year. By the end of the year, his account is worth \$33,000. The time-weighted return for the year is closest to: A) 7.0%. B) 5.5%. C) 10.4%. View Answer and Explanation January – March return = 51,000 / 50,000 − 1 = 2.00% April – June return = 60,000 / (51,000 + 10,000) − 1 = –1.64% July – December return = 33,000 / (60,000 − 30,000) − 1 = 10.00% Time-weighted return = [(1 + 0.02)(1 − 0.0164)(1 + 0.10)] − 1 = 0.1036 or 10.36% When figuring the time-weighted at the end of the problem, aren’t you supposed to take the cube root of 10.36%? Can anyone explain why that wasn’t done here like it usually is when using the geometric mean?

If it asked for the time-weighted return, it’s 10.36%. They’re asking for the return for the entire holding period.

They might ask for the monthly, time-weighted return, for example; you’d calculate (1.1036)^(1/12) –1 = 0.82%.

There’s not much reason to take the cube root: note that the periods of time for the three returns aren’t equal.